Can someone explain the layer system?
Layers



Layers are a way of splitting up and organizing the objects in your scene.  

In the 3D viewport.

You can select the currently visible layers in the footer of the 3D view or under properties panel > render layers tab > layer rollout > scene layer selector.  (With any layer selector you can hold down Shift to select multiple layers.)


You can see what layers have objects on them by the dots in the layers' boxes, layers containing selected objects will have an orange dot.  You can move the slected objects to a new layer by pressing M and selecting the layers you want in the pop-up layer selector.

In the scene section of the properties Window, there are four sets of layer boxes that control which layers are computed in the render and how:




Scene:  The scene layers determine two things: which layers are shown in the 3D view, and which layers are taken into account in the render.  (As mentioned earlier this layer selector parallels the one in the footer of the 3D view.)
Layer:  The layer layers determine which layers are visible to the camera in the render.  This allows you to have certain layers counted into the render (by selecting them under scene), but not visible to the camera (by not selecting them here).  
Exclude:  Any layers selected under exclude will not be calculated in the render in any way.
Mask:  Objects in layers selected under mask will be rendered as transparency to the camera.  They will take part in all other calculations if selected under layer though.


Here is an example showing the functions of the 4 different layer selectors.  The ground plane, light, and camera are on layer 1 (top left); and the red box is on layer 2.

Click to enlarge.

Render Passes



Render passes are simply different ways of rendering a scene, often used in the compositor for post processing effects.  You can choose which passes you want to render under properties panel > render layers tab > passes rollout.  Which ever passes you select will then appear as outputs on the render layers node in the compositor.


Render Layers



I think what may be confusing you is that there is also something called render layers.  Render layers are a way of rendering the scene multiple times with different settings (including passes and visible layers).

You can add new render layers and select which ones should be rendered at the top of the render layers tab.